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in how many ways can 9 identical balls be distributed among four baske

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in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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New post 17 Mar 2018, 03:28
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Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126
Please anybody solve this question and explain me how did you solve this question i am not able to do this question thanks in advance
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in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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New post 17 Mar 2018, 05:08
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rishabhmishra wrote:
Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126
Please anybody solve this question and explain me how did you solve this question i am not able to do this question thanks in advance


This question is based on Distribution principle. It can be done in two ways

This is like find total Natural number solution of an equation a+b+c+d = 9

Method 1:

Natural Number solution of an equation in r variables = (n-1)C(r-1) = (9-1)C(4-1) = 8C3 = 56

Method 2:

Assign one ball to each basket i.e. 4 balls are gone and we are left with only 5 balls to assign to the baskets
Now every basket needs balls from 0 to 5
a+b+c+d = 5
and a, b, c and d may be any integer from 0 to 5
Now manually find solutions which is long but you get a pattern in that as well leading you to the answer i,e, 56

Answer: option B
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Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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New post 17 Mar 2018, 12:20
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rishabhmishra wrote:
Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126
Please anybody solve this question and explain me how did you solve this question i am not able to do this question thanks in advance


If we were to manually list down all the possibilities for the four baskets(and the 4 digit number is the total number of balls in the basket number one, two, three, and four)

6-1-1-1 (4 ways) - 6111,1611,1161,1116
3-2-2-2 (4 ways) - 3222,2322,2232,2223
4-2-2-1 (12 ways) - 4221,4212,4122,2124,2142,2421,2412,2241,2214,1224,1242,1422
4-3-1-1 (12 ways) - 4311,4131,4113,3411,3114,3141,1341,1431,1143,1134,1413,1314
1-2-3-3 (12 ways) - 1233,1323,1332,2133,2331,2313,3321,3231,3123,3132,3312,3213
5-1-1-2 (12 ways) - 5112,5121,5211,2511,2151,2115,1215,1251,1125,1152,1521,1512


Therefore, there are \(4*2 + 12*4\) or 56 ways(Option B) in which the 9 balls can be arranged among the 4 baskets.
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Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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New post 17 Mar 2018, 22:45
rishabhmishra wrote:
Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126
Please anybody solve this question and explain me how did you solve this question i am not able to do this question thanks in advance



hi..

the question does NOT mention about the baskets, if they too are identical..
1) Identical boxes..
a) 6,1,1,1
b) 5,2,1,1
c) 4,3,1,1
d) 4,2,2,1
e) 3,3,2,1
f) 3,2,2,2
so 6 ways

2) non identical boxes..
a) 6,1,1,1 ............. \(\frac{4!}{3!} = 4\)
b) 5,2,1,1............. \(\frac{4!}{2!} = 4*3=12\)
c) 4,3,1,1............. \(\frac{4!}{2!} = 4*3=12\)
d) 4,2,2,1............. \(\frac{4!}{2!} = 4*3=12\)
e) 3,3,2,1............. \(\frac{4!}{2!} = 4*3=12\)
f) 3,2,2,2............. \(\frac{4!}{3!} = 4\)
so 4+12+12+12+12+4=56
In these cases we can take it as an equation a+b+c+d=9, as shown above..

For other type of such possible scenarios
https://gmatclub.com/forum/topic215915.html
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Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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New post 18 Mar 2018, 20:14
chetan2u

If we remove the last condition i.e. such that each basket gets at least one ball. The possible combination - 9!^4 am I correct?
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Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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New post 14 Jun 2019, 11:11
GMATinsight wrote:
rishabhmishra wrote:
Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126
Please anybody solve this question and explain me how did you solve this question i am not able to do this question thanks in advance


This question is based on Distribution principle. It can be done in two ways

This is like find total Natural number solution of an equation a+b+c+d = 9

Method 1:

Natural Number solution of an equation in r variables = (n-1)C(r-1) = (9-1)C(4-1) = 8C3 = 56

Method 2:

Assign one ball to each basket i.e. 4 balls are gone and we are left with only 5 balls to assign to the baskets
Now every basket needs balls from 0 to 5
a+b+c+d = 5
and a, b, c and d may be any integer from 0 to 5
Now manually find solutions which is long but you get a pattern in that as well leading you to the answer i,e, 56

Answer: option B



Could you please explain what "natural number solution" means?

How would the approach change if the numbers changed?

e.g. if 10 identical balls were distributed among 4 boxes and each box gets at least 2 balls, in how many ways could the balls be distributed?

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Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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New post 16 Aug 2019, 11:14
rishabhmishra wrote:
Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126
Please anybody solve this question and explain me how did you solve this question i am not able to do this question thanks in advance


As Baskets are 4, solution should be a multiple of 4. Instead of calculating it the long way, we can use Triage in this question.

Only solution multiple of 4 is B. Hence the Answer.
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Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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New post 16 Aug 2019, 11:25
Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126

Correct Answere is B
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Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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New post 18 Aug 2019, 23:07
Anuragjn reasoning seems wrong to me, as we are summing different numbers of 4 baskets (not multiplying), so not necessarily divisible by 4

Can someone please explain?
Is this question within scope of gmat?
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Re: in how many ways can 9 identical balls be distributed among four baske   [#permalink] 18 Aug 2019, 23:07
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